We evaluate the tracking performance of the filter via the root-mean-squared error (RMSE), defined at any assimilation step \(k\)  as \(\text{RMSE}_k \Vert \bar{z}_{k} - z^*_{k}\Vert _{2}/\sqrt{n}\) . To measure the concentration of the ensemble members, we compute the ensemble spread,
defined at any assimilation step \(k\)  as \([\text{tr}(\widehat{\Sigma }_{k})/n]^{1/2}\) , where \(\widehat{\Sigma }_{k}\)  denotes the ensemble covariance matrix
at step \(k\) .
In Sections REF –REF , we report the time average of each of these metrics (e.g., over the 2000 test assimilation steps) as a function of the ensemble size \(M\) .
In addition, we compute the coverage probability of the intervals given
by the empirical \(2.5\%\)  and \(97.5\%\)  quantiles of each marginal of
the ensemble, i.e., the frequency with which the \(i\) th component of
\(z^*_{k}\)  is contained in the \(i\) th marginal interval.
For each component,
this frequency is of the form \(k/T\)  where \(k \in \lbrace 0,\dots ,T\rbrace \)  denotes the number of intervals that contain the corresponding component of the true state, and
\(T\)  represents the total number of assimilation times.
We then average these coverage probabilities over the \(n\)  components
of the state.
Lastly, we also evaluate the quality of the ensemble by computing the continuous ranked probability score (CRPS) [1]}, [2]}. The CRPS for the \(i\) th marginal component of the state at step \(k\)  compares the ensemble's empirical CDF \(\widehat{F}_{k,i}(z)\)  with a Heaviside function centered at the true state, \(H(z - (z^{*}_{k})_{i})\) , as given by \(\text{CRPS}_{k,i} = \int (\widehat{F}_{k,i}(z) - \mathbb {1}((z^{*}_{k})_{i} \le z))^2 dz\) . We average \(\text{CRPS}_{k,i}\)  over state components \(i\)  and assimilation steps \(k\) .
